Dynamics of a Rotor-pendulum with a Small, Stiff Propeller in Wind

As small rotorcraft grow in capability, the possibilities for their application increase dramatically. Many of these new applications require stable outdoor flight, necessitating a closer look at the aerodynamic response of the aircraft in windy environments. This paper develops the equations of motion for a single-propeller test stand by analyzing the blade-flapping response of a small-stiff propeller in wind. The system dynamics are simulated to show behavior under various wind conditions, and stable system equilibria are identified. Experiments with a rotor-pendulum validate the simulations, including system equilibria and gust response. INTRODUCTION Small unmanned aerial systems (UAS) are transforming from hobbyist entertainment into utilitarian machines. UAS have been tasked with objectives such as surveying farmland and aiding in natural disasters [1] that require multi-rotor aircraft to fly outdoors in potentially adverse weather. High winds pose a great challenge to small UAS [2–4], and developing an understanding of how they respond to wind and the mechanics behind that response is key to compensating for them. This paper uses a combination of theoretical and experimental work to describe the forces and moments experienced by a single quadrotor-type propeller in a uniform wind. Experimental results are collected with a test stand in wind using a Gemfan 5030 propeller commonly used on quadrotor helicopters. These results promise to ∗Address all correspondence to this author. improve guidance and control algorithms for small, multi-rotor helicopters. Although an important part of full-sized helicopter dynamics, blade-flapping is often assumed to be negligible in small quadrotor vehicles [5, 6]. For indoor flight with relatively small advance ratios, this assumption has proved valid even for highly aerobatic flight [7, 8]. However, for outdoor flight in wind, the effect of blade flapping and other aerodynamic phenomena must be re-evaluated [4]. When a helicopter rotor moves forward in air, the advancing side of the rotor produces more lift than the retreating side, which causes a roll moment on the blades [9]. Many studies [10–13] indicate that this moment causes the rotor blades to react with a maximum deflection at 90◦ phase delay, i.e., above the helicopter’s nose, due either to a gyroscopic effect or the blade frequency response. Others provide the equations for flapping without explicitly indicating the expected phase delay [14, 15]. This paper provides a more comprehensive look at the blade-flap dynamics of a small, stiff propeller commonly used in small UAS. Hoffmann et al. [12] and Yeo et al. [16] each measured a quadrotor propeller response to wind. Hoffmann et al. [12] tested a single propeller in wind to identify the flap angle, and showed that the hub experiences a force in the direction of the wind. Yeo et al. [16] tested a single-degree-of-freedom pitch stand with two propellers, as well as a fixed, rigid propeller in an edgewise flow, and found that forces and moments scale with free-stream velocity as suggested in [12]. This paper investigates the source of the forces and moments on a single propeller in wind, and describes blade-flapping dy1 Copyright c © 2016 by ASME namics based on first-principle analyses. A simplified set of analytically tractable equations predict the phase delay and flap amplitude of a small, stiff propeller, the results of which are compared to experimental data. The experimental testbed consists of a two-degree-of-freedom rotor-pendulum, which is a spherical pendulum affixed with a spinning propeller. The long arm of the spherical pendulum increases the effect of the hub forces, demonstrating the propeller’s response to wind. The contributions of this paper are (1) a detailed analysis of the blade-flapping response of a small, stiff propeller in uniform wind, yielding the derivation and solution of the equations of motion for blade flapping and the rotor-pendulum system from first-principles; (2) comparison of the first-principles model to existing experimental measurements of forces and moments at the hub of a propeller fixed in a uniform wind; and (3) the design, fabrication, and testing of a rotor-pendulum test stand that demonstrates the effect of wind on a single propeller. This work increases the theoretical and physical understanding of a small, stiff propeller’s response to wind, which has the potential to yield improved flight stability for small multi-rotor helicopters in adverse weather conditions by virtue of an improved feedback response using flow sensing and control [16]. The outline of the paper is as follows. The first section describes the rotor-pendulum system and develops the equations of motion for a static rotor for comparison to pre-existing experimental data. The second section investigates aerodynamic forces acting on the propeller and derives the equations of motion for the full rotor-pendulum system. The third section provides new experimental results for the rotor-pendulum, with a comparison to model predictions. The final section summarizes the paper and ongoing work. ROTOR DYNAMICS This paper utilizes a rotor-pendulum to investigate the effect of wind on a small, stiff propeller. The rotor-pendulum is a variation of the gyro-pendulum, which is a spherical pendulum with a rapidly spinning mass on the mobile end that causes the system to precess and nutate. Figure 1 shows the rotorpendulum system: a gyro-pendulum with the spinning mass replaced by a propeller. Consider inertial frame I , (O,e1,e2,e3) and intermediate frame A , (O,a1,a2,a3), where a3 = e3 and a1 · e1 = cosθ . Spherical frame B , (O,b1,b2,b3) satisfies b2 = a2 and b1 ·a1 = cosφ . The hub frame is C , (O,c1,c2,c3), where c3 = b3 and c1 ·b1 = cosψ . Let Nb represent the number of propeller blades and the superscript (n), where n = 1,2, or 3, denote the blade index, so that frame D(n) , (H(n),d 1 ,d (n) 2 ,d (n) 3 ) has origin at the blade hinge, and rotates about c2 by the flapangle β . (The blade index (n) is included only where needed for clarity.) Let r denote the displacement along the length of the blade of a point P with respect to O′, and dr be the differential position. The differential forces, moments, and mass are denoted Fdr , dF , Mdr , dM, and mdr , dm, where the quantities F , b 2 = a 2